A Collection of Recommendations
for Reading in the Field of
Mathematics

By Matthew Whitney

How did I get into reading math books for fun? Well, it has to do with why I pursued a mathematics degree at all. My first field was Business, graduating with a degree in Accounting and entering the teaching profession in 1983. Due to many factors the Business department at my school suffered a decline, talk began of possible lay-offs, and I worried that I would soon be out of a job. So, I returned to school to get certification in Mathematics to make myself more employable.

During the process of studying mathematics as a continuing education student during nights and summers, I found myself falling in love with the discipline.>>>More>>>

The following recommendations are grouped alphabetically by title. They are not ranked nor rated in any order.

A-E | F-J | K-O | P-T | U-Z

or

1. A Mathematician's Apology - G. H. Hardy
2. The Art of Mathematics - Jerry P. King
3. Beyond Einstein - Michio Kaku and Jennifer Thomson
4. Beyond Numeracy - John Allen Paulos
5. Billions & Billions - Carl Sagan
6. The Blind Watchmaker - Richard Dawkins
7. A Brief History of Time - Stephen W. Hawking
8. Calculus Gems - George F Simmons
9. Calculus Made Easy - SilvanusThompson and Martin Gardner.
10. The Cartoon Guide to Statistics - Larry Gonick & Woollcott Smith
11. Concepts of Modern Mathematics - Ian Stewart
12. Cosmos - Carl Sagan
13. e: The Story of a Number - Eli Maor
14. Euler: The Master of Us All - William Dunham
15. Experiencing Geometry on Plane and Sphere - David W Henderson
16. Fermat's Last Theorem - Amir D. Aczel
17. Five Golden Rules - John L. Casti
18. Flatland: A Romance of Many Dimensions - Edwin A. Abbott
19. The Fourth Dimension - Rudy Rucker
20. From Here to Infinity - Ian Stewart
21. Geometry, Relativity, and the Fourth Dimension - Rudolf v. B. Rucker
22. The Golden Ratio - Mario Livio
23. A History of Pi - Petr Beckmann
24. How to Solve It - G. Polya
25. Hyperspace - Michio Kaku
26. Innumeracy: Mathematical Illiteracy and its Consequences - John Allen Paulos
27. Islands of Truth: A Mathematical Mystery Cruise - Ivars Peterson
28. Journey Through Genius - William Dunham
29. The Language of Mathematics - Keith Devlin
30. Learning Mathematics for a New Century: NCTM 2000 Yearbook - Edited by Maurice J. Burke
31. The Man Who Knew Infinity - Robert Kanigel
32. The Math Gene - Keith Devlin
33. The Mathematical Experience - Philip J. Davis and Reuben Hersh
34. Mathematical Mysteries - Calvin C. Clawson
35. Mathematical Sorcery - Calvin C. Clawson
36. The Mathematical Universe - William Dunham
37. Mathematics: The New Golden Age - Keith Devlin
38. Mathematics and the Physical World - Morris Kline
39. Mathematics for the Curious - Peter M. Higgins
40. Mathematics - From the Birth of Numbers - Jan Gullberg
41. The Moment of Proof - Donald C. Benson
42. The Mystery of the Aleph - Amir D. Aczel
43. Nature's Numbers - Ian Stewart
44. The Number Devil - Hans Magnus Enzensberger
45. Once Upon A Number - John Allen Paulos
46. Poetry of the Universe - Robert Osserman
47. Prime Obsession - John Derbyshire
48. Principles and Standards for School Mathematics - National Council of Teachers of Mathematics
49. Probability 1: Why There Must Be Intelligent Life in the Universe - Amir D. Aczel
50. Proof - A Play by David Auburn
51. Q.E.D. - Burkard Polster
52. Science Matters - Robert M. Hazen & James Trefil
53. Sphereland - Dionys Burger
54. The Story of Mathematics - Richard Mankiewicz
55. The Story of Mathematics - Lloyd Motz and Jefferson Hane Weaver
56. To Infinity and Beyond - Eli Maor
57. A Tour of the Calculus - David Berlinski
58. Uncle Petros & Goldbach's Conjecture - Apostoios Doxiadis
59. The Universe and the Teacup - K. C. Cole
60. Visions: How Science Will Revolutionize the 21st Century - Michio Kaku
61. What is Mathematics? - Richard Courant and Herbert Robbins (Revised by Ian Stewart)
62. Why Numbers Count - Lynn Arthur Steen (editor)
63. Zero: The Biography of a Dangerous Idea - Charles Seife

A Mathematician's Apology by G. H. Hardy [with Forward by C. P. Snow]. Cambridge: Cambridge University Press, First edition 1940, with forward 1967 (Canto paperback edition, 2001 reprint).

Hardy's "Mathematician's Apology" is world famous in mathematics. G. H. Hardy was one of the greats in mathematics at the start of the 20th century and is perhaps most famous now for having discovered the Indian mathematical genius Ramanujan. This slim volume (153 pages in my edition, of which the first 58 are title pages, etc, and then Snow's introduction) is a worthy read. Some of the most famous quotes about mathematics have been gleaned from this volume. It is an interesting look at how a mathematician viewed his life and his justification of the merits of his work. [I debated on the alphabetizing of this entry and opted to use "A" rather than "M" since the M's are already so crowded. Further, this work is perhaps best known by the three word title, so I feel the emphasis is justified.]
 

The Art of Mathematics by Jerry P. King. New York: Fawcett Columbine, 1992 (Ballantine Books paperback edition, 1993).

This work takes an interesting look at mathematics from an aesthetic point of view. Those of us who love mathematics instantly see its grace and beauty but far too many look on mathematics as ugly and loathsome. The author directly addresses this division. The reason, he contends, is the poor state of mathematics education. King claims that the mathematics community is a cloistered, monk-like, society where the practitioners of math hold themselves above and apart from everyone else. They feel no need to bridge the gap between those who know mathematics and those who do not. This is most true, says King, at the college level where the professors are interested in their own research and teaching is seen as an unfortunate burden that comes with the privilege of being a research mathematician. This is a scathing accusation. [Fortunately, I have been lucky to have worked with a lot of great professors, with very few exceptions.] It is interesting to note that the publishing date coincides with the stirrings of major reforms in math education, particularly calculus reform and the NCTM's revision of their principles and standards. As for my general feelings for this book, I had hoped to find more discussion of what makes mathematics great art. There is much of that, but the book is primarily an inditement of mathematics teaching. I still found it worthy enough to recommend it as a resource for my school's International Baccalaureate program "Theory of Knowledge" teacher.
 

Beyond Einstein: The Cosmic Quest for the Theory of the Universe by Michio Kaku and Jennifer Thomson. New York: Anchor Books, 1995 (revised & updated edition, original published 1987 by Bantum Books).

Cosmology - the study of the creation of the universe - is a major interest of mine. This book I picked up due to my enjoyment of the lead author's later book, Hyperspace. This was a good read - though no where nearly as good as Hyperspace, which was both more informative and easier for the non-expert (like me!).
 

Beyond Numeracy: Ruminations of a Numbers Man by John Allen Paulos. New York: Vintage Books, 1991.

This was the follow-up book to Innumeracy - a fabulous book on the dangers of math illiteracy. That book is MUST reading for any math teacher or a college bound student hoping to study mathematics. This follow-up was more of the same but it did not match the sparkling insights that the first book had. If you enjoy Innumeracy, and must have more ... this book is here. [But just because I didn't like this book as much as the first did not slow me from getting the author's Once Upon a Number.]

Billions & Billions: Thoughts on Life and Death at the Brink of the Millennium by Carl Sagan. New York: Random House, 1997.

This book was a collection of essay's, complied after his all-to-early death. Like his landmark work Cosmos, this is more a look at all of science and critical thinking than it is a math book. Dr. Sagan remains such a hero to me, though, that I had to include him in this listing. Also, this book is the source of one of my math assignments - where I require studetnts to calculate their age -- in seconds. I was inspired to do this after reading a segment where Sagan commented how many seconds it would take to count a billion (counting one number per second). Fascinating!

The Blind Watchmaker: Why The Evidence of Evolution Reveals a Universe Without Design by Richard Dawkins. New York: W.W. Norton & Co, 1996 (originally published 1986).

I picked up this book because I wanted to read more on the the subject of Evolution and I felt it would give some needed background to help me prepare for my grad school program (which was to have a strong science component). To my surprise this book was not only a good read -- but a very mathematical approach to biology. This would make a great read for any prospective science major or math student who is interested in math applications.

A Brief History of Time: From the Big Bang to Black Holes by Stephen W. Hawking. Toronto: Bantum Books, 1988.

Stephen Hawking is a modern legend in his own time. Dispite tremendous physical ailments that have convined him to a wheel chair and only able to speak via a complicated computer voice synthesizer, he is considered by many to be a genius along the lines of Isaac Newton and Albert Einstein. The story of his life is absolutely amazing and this book spent a great deal of time on the "Best Sellers" list. I purchased it not only because of my interest in the subject (cosmology ... the creation of the universe) but also because it contained an introduction by my favorite scientist Carl Sagan. Even though this book crossed over into "popular" culture, it is still a difficult read. This should not be your starting point on the subject. There was a companion film, created some years after the book, which is quite interesting.
 

Calculus Gems: Brief Lives and Memorable Mathematics by George F. Simmons. New York: McGraw-Hill, Inc., 1992.

Calculus Gems combines a series of 33 biographies of great mathematicians (through the 19th century) with 26 explorations into some of the most interesting mathematical ideas that the human species has discovered. The short biographical sketches are excellent and are well within the scope of high school readers. The "memorable mathematics" section requires a more experienced background and quickly went over my head. Most of the ideas are from advanced calculus and that section would have taken me weeks (months?) of serious study to be able to understand at even a limited level. [Instead, I skimmed several of the ideas, though the book will go into my personal library of resource books.] It is no surprise that I was directed toward this work by discovering it on the syllabus for a college course taught by my Analysis instructor at Westfield State College. The biographies are accessible to all and the richness of the mathematics make the book an excellent resource for serious mathematics students.
 

Calculus Made Easy by SilvanusThompson and Martin Gardner. New York: St. Martin's Press, 1998.

This "text" is a revised edition of a book originally written in 1910 by S. Thompson and was revised an updated by M. Gardner. It came highly recommended as an excellent source to learn Calculus from a practical point of view. I found it very informative and clear. It certainly added to my understanding of the Calculus.

The Cartoon Guide to Statistics by Larry Gonick and Woollcott Smith. New York: HarperPerennial, 1993.

I needed to learn about a topic called t-Tests, so I checked this book out of the school library, hoping for a quick refresher on a topic I have not formally studied in over 20 years. While it did not entirely get me up to speed on the topic I was specifically interested in, it did provide an interesting overview of the general topic of statistics. While I would not recommend the work as a resource for studying for a statistics final, it does provide the reader with an interesting "big picture" of the field.
 

| A-E | F-J |
| K-O | P-T |
| U-Z |

Top

Concepts of Modern Mathematics by Ian Stewart. New York: Dover Publications, Inc, 1995 (reprint of 1981 edition, originally published in 1975)

This reprint edition was the author's attempt to explain much of the basics underlying the "new math" philosophy (called "modern math" in the author's England) that was such a controversial topic in the 60's and 70's. Having been a student who was one of the guinea pigs for "new math," I found the topic to be of great interest. This allowed me to see some of the reasons for why I was taught much of the math I learned in my early years. I think the book has some merits for being a collection of a wide range of topics, but it will not light many fires of imagination. I have read several articles by the author as well as at least one other full length book, and I feel his style has grown considerably since this early work. By the end of the book I found myself skipping over sections that were of little interest. [See references to Ian Stewart in my authors search page.]

Cosmos by Carl Sagan. New York: Random House, 1979

Not really a "math" book -- more a book on astronomy and general science, but it stands as one of the most inspirational books I've ever read. In this companion book to the breathtaking TV series Dr. Sagan takes the reader on a tour of the entire Cosmos / universe - from its birth to present day. This book, read long before I got hooked on math had a profound impact on me and I cannot recommend it too highly.

e: The Story of a Number by Eli Maor. New Jersey: Princeton University Press, 1994

Definitely a book for "math folk." The number e ranks up there with the likes of "pi" among the most important numbers you will encounter in your daily travels. What's that ... you've never heard of e? For shame on your math teachers. You'll be surprised how common e (approximately equal to 2.71828128 ... with no repeating pattern and an endless decimal expansion) really is. It is vital to calculate many bank interest rate problems, to name probably just the most dull example. In reading this book, though, I did find how far I still need to go in my own math education. It was an entertaining read, though, and I heartily recommend it for all serious math students.

 Euler: The Master of Us All by William Dunham. The Mathematical Association of America, 1999

Euler (pronounced "oiler") is one of the giants of the Pantheon of great mathematicians. He lived in the 1700's and contributed in nearly every field of mathematics being studied at the time. He was one of the first "modern" mathematicians and will certainly be the last to achieve such success in so many diverse fields. Today, mathematics is so highly specialized and compartmentalized that greatness in more than one or two fields is nearly unimaginable. Euler was a master in nearly every field. He wrote new, original and highly significant mathematics at the rate of a newspaper columnist. Author William Dunham has collected several of his most important discoveries / contributions from eight different fields. In the book, Dunham presents the background on the problem / theory being discussed, then "enters Euler," where he explains Euler's findings and methods, with each followed by an epilogue of how Euler's work influenced others. This humble reader was more than a little lost in the blizzard of algebraic gymnastics. It would have taken me weeks of determined study to draw from the book all that the author intended. Instead, I focused on the fields where I have had some experience and in the others just gave a nod of appreciation to this legendary genius. The book will be a superior resource in my library and I have already loaned a photocopied chapter to a student working on an "extended essay" on the Euler Identity [e^{p i} + 1 = 0]. College professors should definitely include this book in an advanced course that surveys math history.
 

Experiencing Geometry on Plane and Sphere by David W Henderson. New Jersey: Prentice Hall, 1986

Again I am recommending a text book which I have used in college. This text is unique in that I used it both as an undergraduate with Dr. Maureen Bardwell at Westfield State College and with Dr. R. Daniel Hurwitz (Skidmore College) in my Rensselaer Polytechnic Institute master's program. The text is a slim volume, filled more with questions to explore than a series of facts that you will need to memorize. The book forces you to think by challenging virtually all preconceptions you hold regarding geometry (and the structure of the world around you). You will find even your most basic notions challenged. The end result is that you shall be better versed in communicating your geometry knowledge. I cannot envision a geometry education that does not incorporate this approach at least to some level. This book is highly recommended!

Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem by Amir D. Aczel. New York: Four Walls Eight Windows, 1996.

There are several books out on the subject of Andrew Wiles' proof of Fermat's Last Theorem, and this was one I read during my first summer at RPI. While much of the math went sailing over my head, I enjoyed getting a feel of what was happening at the top of the math ladder. At the time I was studying math at a level I had never dreamt of, so that was quite fun. That summer I wrote my term paper on the Collatz Conjecture (the 3N + 1 problem), which is regarded as a modern successor to FLT ... so I felt quite connected to the subject. The author, Amir D. Aczel, has several works on this list and I highly recommend his work.

 Five Golden Rules: Great Theorems of 20th Century Mathematics - and Why They Matter by John L. Casti. New York: John Wiley & Sons, Inc., 1996.

This work was slightly disappointing since I feel the author did not successfully introduce the topics to those who are not already deeply into the subjects. The goal of the work was to disseminate what is going on in modern mathematics, and on that level it was somewhat successful. Too often students think that mathematicians sit in musty offices with compasses bisecting angles and factoring trinomials. Hardly! Casti's book is a decent look at what is really being studied by today's mathematicians. I do like that it shows that new math is currently being developed, but it was a bit discouraging that I did not feel I was doing more than scratching the surface. Still, the book has worth to a college student looking beyond the the next course in his or her major.
 

Flatland: A Romance of Many Dimensions by Edwin A. Abbott. New York: Signet Classic edition 1984 (originally published 1884).

This is a somewhat controversal book. What? Controversy and math ... over what? The author intended (I feel) to write a both a satire on Victorian English society as well as to make an argument for the possible existence of a divine being. The latter he accomplished by posing the problem of how we three dimensional beings can comprehend the fourth physical dimension. He did this through an allegory of a two-dimensional being encountering a three-dimensional world. Thus, the book becomes a great mathematical exploration of geometry of in terms of dimensions. The satire on the class system of English society, with particular reference to the role of women, can be misinterpreted, I feel. One needs to read the book remembering when it was written and in what spirit. Discrimination of women from participating in mathematics has occurred, but this book should not be overlooked because it may be mis-read in those terms. I firmly believe the author hoped to breakdown barriers, and viewed in that light I feel the book has a great deal of merit.

An amusing post script to this recommendation is that one of my colleagues in my Master's degree program was thumbing through a copy of Flatland and came upon this passage (the opeing lines of Chapter 13):

"It was the last day but one of the 1999th year of our era, and the first day of the Long Vacation. Having amused myself till a late hour with my favourite recreation of Geometry, I had retired to rest with an unsolved problem in my mind. In the night I had a dream."

This occurred in late July 1999, just days before we were to complete our MS degrees, about to start our "long vacation." And the course that absorbed the bulk of our studies that summer was Prof. Dan Hurwitz' Geometry course! (And that was a wonderful course!)

[See the "sequel" - Sphereland.]

The Fourth Dimension: A Guided Tour of the Higher Universes by Rudy Rucker. Boston: Houghton Mifflin Company, 1984.

I got this book on the strength of the author's Geometry, Relativity & the Fourth Dimension. While I enjoyed both, I feel the author did a better job on the topic with the aforementioned work. Rucker has a fun style, though, and the illustrations by David Povilaitis were quite fun, so this books is worth looking for.

From Here to Infinity: A Guide to Today's Mathematics by Ian Stewart. Oxford: Oxford University Press, 1996.

This is another math book that I grabbed thinking it would help me in my Master's degree courses. A tip of the cap to Westfield State College's Julian Fleron, Ph.D. who really got me thinking about modern developments in math. While I am by no stretch of the imagination a mathematician on the cutting edge of the science, as a math teacher I try to get my high school students to realize that all the math that is out there was not established thousands of years ago by some dead Greek dudes. New fields are opening up, discoveries are being made, and there is original work to be done by people living and working today. One of the neat things the author highlights in this book is the large amount of things that are cited as being discovered / written about after 1983 (the year of my first undergraduate degree). Stuff from AFTER I got out of college! The book is worth a look if you are serious about math (but don't get discouraged by a good deal of it that might go over your head ... the level is quite high).

Geometry, Relativity, and the Fourth Dimension by Rudolf v. B. Rucker. New York: Dover Publications, 1977.

A primary source for my non-Euclidean geometry paper in Geometry II, it was also my first serious excursion into higher-dimensions. This has become a topic of serious interest to me and I became quite a fan of Rudy Rucker's writing. I rate this an an excellent book on the geometry of higher dimension and a great first book to read on the topic.

 The Golden Ratio: The Story of Phi, The World's Most Astonishing Number by Mario Livio. New York: Broadway Books, 2002.

The Number Phi (F) is not as famous to non-mathematicians as is Pi (p), but it certainly ranks as one of the most bizarre numbers that exist. Phi first was studied as a ratio of sides or rectangles or triangles of a certain relationship. Take a rectangle of length (x + 1) and width x. The ratio of (x + 1) to x is the same as the ratio of x to 1. This relationship occurred often in the Platonic solids and in other regular polygons studied by the ancient Greeks and something mystical came to be associated with the number to which these ratios were equal (the first few decimals are 1.61803...). In this book I learned that we can consider phi to be the "most" irrational of all numbers for its unique property of being equal to a couple of amazing continued numbers. Phi is the limit of the series: the square root of (1 + the square root of (1 + the square root of (1 + the square root of (1 + the square root of ( ... continued infinitely far. Phi is also equal to the continued fraction: 1 + 1 divided by (1 + 1 divided by (1 + 1 divided by (1 + ... again, continued infinitely far. That is very startling, but mathematics has a tendency to surprise us. As a general mathematics book, this volume works fairly well. I was particularly impressed with the author's willingness to reject claims that phi showed up just about everywhere. Many places people claim to find phi (in art or architecture) seem to be mere coincidence or number juggling by those looking to find something, rather than intentional design by the artist / architect.
 

A History of Pi by Petr Beckmann. New York: Barnes and Noble Books, 1993 (Originally published by The Golem Press, 1971)

This book is one of the most delightful math books I've read. All you every wanted to know about Pi (p) is in there. It is amazing how such a trivial topic could have so much depth. Math folk will love this -- and it is a surprisingly easy read (so much so that I have recommended it as required reading for the HS students in my school's honors programs).

How to Solve It: A New Aspect of Mathematical Method by G. Polya. Princeton, NJ: Princeton University Press, 1957 (Second Edition).

How to Solve It is a fairly famous math book. The author uses it as a forum to advocate a more "problem solving" approach to mathematics education, rather than the cold, "drill & kill" repetitive style of teaching that characterizes poor math teaching. While citing that there is a place in math education for some degree of practice work, the author fervently puts forth the hypothesis that students will learn best if they learn through discovery and that students can be taught methods to problem solve. The work was originally written in 1945, and it seems math education has changed much since that time. Through agencies such as the National Council of Teachers of Mathematics, a more engaging, problem-solving approach has become more of the norm than the exception. As a reader, I found some valuable advice, which would be particularly useful to a new teacher, but I would not recommend "rushing out" to read this one. On a somewhat perplexing note, as "forward thinking" as the book is, I do not recall the author ever to have referred to a mathematician, mathematics student, nor a mathematics teacher as female. This is a sad condemnation of the times Polya lived and worked, when gender discrimination was virtually a matter of course.

 Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension by Michio Kaku. New York: Anchor Books, 1995 (originally published by Oxford U. Press, 1994).

This math/quantum physics book is a better mind trip than the best science fiction! It introduces the reader to the real possibility of the nature of the universe as being made up of higher dimensional things called "superstrings." I was utterly enthralled and just could not put it down. It made a great source for my Geometry II paper on non-Euclidean geometry. It is not a simple read, but if you enjoy mainstream astronomy or cosmology books, this should be no trouble. A great read! The author, Michio Kaku, has a real gift for making extremely difficult topics very understandable.

Innumeracy: Mathematical Illiteracy and its Consequences by John Allen Paulos. New York: Vintage Books, 1988.

This is must read for math teachers! The author takes you on a tour of the dangers of math illiteracy in our society. Much of the style is the presentation of a topic - many from probability - followed by an exploration of the many ways peole misunderstand the math involved. Much of the math is simple statistics - not high level calculus. I expect most math teachers will give a second thought on the emphasis placed on various topics after reading this book. In addition to its cautionary value, there is still a wealth of fascinating math in its pages (for example, it is here I first encountered the equation: e^{p i} +1 = 0). That knocked my socks off when I first saw it ... all the five "big" constants in math (e, p, i, 1, and 0), tied into one equation. Too cool!

Islands of Truth: A Mathematical Mystery Cruise by Ivars Peterson. New York: W.H. Freeman and Company, 1990.

This was the second book required for my "Senior Seminar in Math" at Westfield State College. It was the author's follow-up to The Mathematical Tourist, but it seemed to be just more of the same. The sparkling insights of the first book were not matched in this work, though it was not necessarily a poor attempt. If pressed for a recommendation, either book is worthy, but I'd advise against reading both back-to-back (a sentiment echoed by my classmates).

Journey Through Genius: The Great Theorems of Mathematics by William Dunham. New York: Penguin Books, 1991 (originally published by John Wiley & Sons, 1990).

Easily one of the GREATEST math books I have had the pleasure to read. This book caused me to fall in love with proofs. I only wish I had read it when I was struggling to learn how to write proofs. (I still am not great at that, but I appreciate the beauty and necessity of proofs far more now than before.) I discovered it thanks to my partner in a geometry research project (on Eratosthenes' estimation of the circumference of the Earth in 240 BC), but did not read it in total until some time later. As the subtitle implies, Dunham takes the reader on a tour of the great theorems of math, both in their historical context and from a strict mathematical point of view. The sense of awe and delight in the mathematics is of such a high level that those who love mathematics will be in pure heaven reading this book. This is writing about mathematics that should stand as the benchmark for all others to aspire to .

| A-E | F-J |
| K-O | P-T |
| U-Z |

Top

The Language of Mathematics: Making the Invisible Visible by Keith Devlin, New York: W. H. Freeman and Company, 1998 (paperback edition, 2000).

In this work the author attempts to paint on a large canvass how mathematics impacts our lives. As with many of the math books on this list, this work is geared toward inspiring the reader toward a deeper appreciation of mathematics. Devlin is generally very successful in this endeavor. I particularly enjoyed the chapter on the development of the calculus (which was my prime teaching focus at the time I read this). At times the author bogs down into details, but for the most part the writing is crisp and engaging. This work would make a suitable resource for an undergraduate math student to use as a spring board for deeper studies.
 

Learning Mathematics for a New Century: NCTM 2000 Yearbook Edited by Maurice J. Burke, (General Yearbook Editor) Frances R. Curcio. Reston, VA: National Council of Teachers of Mathematics, 2000.

This is an interesting collection of essays directed at the way mathematics education has changed and in what directions it should take to improve as we enter the twenty-first century. The quality ranges from excellent to adequate, but leans toward the high end. Teachers should find this quite engaging. The essays are grouped in the following themes: (1) Numeracy and Standards, (2) Technology, (3) the School Mathematics Curriculum, and (4) Improving Mathematical Learning Environments.
 

The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel. New York: Washington Square Press, 1991.

This is a "must read" for every aspiring mathematician (and for anyone who enjoys a fascinating biography). Ramanujan was a mostly self-schooled mathematician who rose out of obscurity in southern India around 1900. He contacted several prominent English mathematicians, seeking validation of his work and assistance in getting the work published. G. H. Hardy recognized that Ramanujan was a genius of the highest order and convinced him to travel to England. Hardy helped fill in the gaps in formal training that Ramanujan lacked while not getting in the way of his intuitive methods and revolutionary genius. Tragically, Ramanujan did not adapt well to the cold northern climate, contracted tuberculosis, and died shortly after returning to India at a young age. We are left to imagine what heights he could have reached if his potential had been seen earlier. The biography by Kanigel paints exceptionally vivid portraits of Ramanujan and Hardy as well as exploring the world of Ramanujan's native India and the university town of Cambridge. This is a truly remarkable read, both for the dramatic life of this tragic genius and for the insight into two very different cultures.
 

| A-E | F-J |
| K-O | P-T |
| U-Z |

Top

The Math Gene: How Mathematical Thinking Evolved and Why Numbers are Like Gossip by Keith Devlin. Great Britain: Basic Books, 2000.

The author of this title puts forth an interesting idea, that the same evolutionary trail that led humans to develop language also led to our developing mathematics. In that the author argues that everyone (virtually) has the attributes that can be developed to allow them to do mathematics at a somewhat sophisticated level. For those students who claim they just do not have what it takes, this book asserts that mathematical ability is already hardwired into each brain. Now, how to move from having the tools to developing the talent is the big question. I would rate this work as highly illuminating, well worth the time of any who is interested in mathematics education. I found it very readable though it bogged down a bit in the details of language structure.

The Mathematical Experience by Philip J. Davis and Reuben Hersh. New York: Mariner Books, 1998 (originally published 1981).

The authors here have collected a variety of stand alone essays on mathematics, clustered into various themes (what is mathematics?, varieties of mathematical experiences, teaching and learning, certainty and fallibility, et al). I found the book to offer an interesting cross section on the mathematics profession. Several of the essays were excellent, though some were less than engaging. This work was well worth my time but I would hold off from giving this a very high recommendation. There are certainly several other works mentioned on these pages that I would urge a new reader in mathematics to explore first.

Mathematical Mysteries: The Beauty and Magic of Numbers by Calvin C. Clawson. New York: Plenum Press, 1996.

Another fabulous volume, easily accessible to the advanced and interested high school student or undergraduate. This is an excellent excursion into Number Theory, which is rapidly becoming my favorite area of math. This book ranks on a par with Journey Through Genius as one of the most enlightening, awe inspiring math books I have stumbled upon and have enjoyed with sheer delight. Some math books get me thinking and deepen my understanding of my discipline and for that I rate them as valuable reads. Some strike a cord and stand above the others as if they were a great Shakespearean drama. They excite me, leaving me breathless as to what I will discover on the next page. This was such a book. I would make this required reading in any Number Theory course.

Mathematical Sorcery: Revealing the Secrets of Numbers by Calvin C. Clawson. New York: Plenum Press, 1999.

This book would make an admirable introduction to some of teh wonders of "higher math." I would particularly recommend it to high school / early college students who are just entering their mathematics studies. It contains many wonderous observations and explorations that I did not encounter for several years into my studies. This is the author's intended audience. In my case, reading it nearly ten years after returning to school to major in mathematics, however, I was at a distinct disadvantage in term of how the work would impact me. Clawson's previous work [Mathematical Mysteries] was geared at a slightly more advanced reader, with far less "primer" material, hence I enjoyed that work more.

The Mathematical Tourist: Snapshots of Modern Mathematics by Ivars Peterson. New York: W. H. Freeman and Co, 1988.

Technically, this was a text book that I was required to read, not one I picked up for pure pleasure - but one I really enjoyed nonetheless. It was required for Prof. Julian Fleron's Senior Seminar in Math at Westfield State College, one of my favorite undergrad math classes. This is a great place to start looking at the many areas of math that are currently being studied at advanced levels. It is not a book filled with equations, but is meant of popularize math. If you love math, you should enjoy this book. [Note, the author's follow-up book Islands of Truth: A Mathematical Mystery Cruise turned out to be a tad disappointing in that it was somewhat "more of the same."]

The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems and Personalities by William Dunham. New York: John Wiley & Sons, 1994.

This book is another example of a less than stellar follow-up to a fantastic effort. The author's Journey Through Genius is a work of genius in its own right, but this book turned out to be a modest step down. But then, how often can one catch lightning in a bottle? Still, the book has many virtues. I enjoyed the alphabetical listing of important ideas and people from math. The format made for nice, quick reads on a topic in brief, never getting bogged down in details. But, the sweep and scope of JTG was not there. It is still a worthy book for your library, and is probably one of the most accessable books for HS students in this list.

Mathematics: The New Golden Age by Keith Devlin. New York: Columbia University Press, 1999.

As the subtitle indicates, this book argues that we are in the midst of a new golden age in the field of mathematics. The author takes the reader on a fairly sophisticated overview of the several areas of mathematics where new arenas of research are flourishing. The argument is quite compelling, laying to rest (?) the misconception that the study of mathematics is nothing more than the memorizing of the work of dead ancients. One of my goals as a math teacher is to awaken students to this very fact ... that new developments and discoveries in math are on the horizon. If there is a problem with this book it is that the level of difficulty of the mathematics is high. A high school or early college reader by necessity will have to skip over many sections (as, indeed, I myself did ... it is a heavy duty book). Still, there is much to be gained by a tour through its pages.

Mathematics and the Physical World by Morris Kline. New York: Dover Publications, Inc, 1981. [Original copyright: 1959]

Math permeates the physical world and the author here did an admirable job at being comprehensive, looking at how math has been used to unlock many of nature's secrets. Unfortunately, the style was not electrifying and the book did not stir me with awe for the wonders of mathematics. The most entertaining bit was a line about how mathematics can fill roles that none ever expected, such as the use of branches of math in the programming of computers. The math involved had been mere intellectual curiosities for centuries with no real utility, but with the advent of computers the math gained practical use. The entertaining bit was the mention of the basic component of the computer ... the radio vacuum tube. After nearly falling out of my chair with laughter I looked up the original copyright date and noted that the book was authored pre-1960.
 

Mathematics for the Curious by Peter M. Higgins. Oxford: Oxford University Press, 1998.

This book was a gem! One of several listed here that covers a spectrum of topics, intended to give the reader a taste of the many branches of math and how looking deeper into these topics presents a delightful course of study. I kept a note pad at my elbow while reading this one so that I could jot down notes of things that I wanted to bring to my classes. The problems ranged from those that were simple to state but deliciously difficult to solve to the more obscure sort of problem one would expect. The presentation reminded me to a great degree of the first math course I took in my MS program at Rensselaer, under Professor Jim Matthews (Siena College) ... challenging math for anyone curious enough to ask questions. Rank this one very high on your list of books to look into.

Mathematics - From the Birth of Numbers by Jan Gullberg. New York: W. W. Norton and Company, 1997.

THE book for easy, thorough reference. This is an emense omnibus of virtually all the math resources you will ever need. It includes detailed descriptions of just about everything you could want - clearly written, info on historical topics, well organized, lots of witty illustrations. It is a must for any mathematician! I have made great use of it, getting a quick tutorial on topics before looking for more detailed information elsewhere or for a clearer explanation of a particularly sticky topic. It is a tad expensive, but it is worth every penny!

The Moment of Proof: Mathematical Epiphanies by Donald C. Benson. New York: Oxford University Press, 1999.

This was a major disappointment. On the surface it looked like it would rank along side Journey Through Genius as a triumph of making proofs accessible and exciting. It failed in both accounts. There are "moments" where the book is enjoyable, but they are few and far between. For my tastes, there was too much attention paid to puzzle solving, a topic I have little interest in. When a really important proof was being discussed, it was either raced through with too little attention, detracted from by numerous side-bar comments, or the details simply were not made clear. Compounding the flaws in the text is an exceptionally poor graphical layout. Illustrations are not clearly linked to their text and the captions' font is nearly identical in format to the regular text. This latter flaw causes the delination between captions and text to be vague and confusing at best. This is one of the very few books on this list where I have skipped over large sections due to my boredom with the material (or at least the presentation).
 

The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity by Amir D. Aczel. New York: Four Walls Eight Windows, 2000.

The Infinite is a topic of great fascination and intrigue. How can something extend infinitely, such as a set of numbers for which there is no largest number, yet there are collections which have even more members? This work explores the discovery of this fact (and much more) and how our understanding of the Infinite evolved, from ancient times up through the achievements of modern Analysis. Principle players in this quest include the great Georg Cantor and Kurt Godel, both who died under tragic circumstances, suffering mental breakdowns. This is as much biographical as it is mathematical and is accessible to any student from first year calculus and up. It is another fine work by author Amir D. Aczel.
 

Nature's Numbers: The Unreal Reality of Mathematics by Ian Stewart. New York: Basic Books, 1995.

Bought while I was at RPI, I began it on the night before my last class in Dynamical Mathematical Systems, and the first passage I read (having flipped it open to a "random" page) related directly to the course. It struck me as a very appropriate way to conclude that course (which dealt with the math of growing, dynamic systems). This is an excellent book to explore if you are considering math as a career or if you are interested in the math that underlies so much of the natural world. Also, the author, Ian Stewart, is among the top writers in the field of mathematics.

The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger with illustrations by Rotraut Susanne Berner, translated by Michael Henry Heim. New York: Henry Holt and Company, 1997 (English tranlastion 1998).

The Number Devil is a children's book about mathematics. It introduces children to some of the great ideas of mathematics by way of a young boy dreaming of a "number devil," an impish character who teaches the boy how to have fun playing with numbers. The style is very reminiscent of Dr. Suess or Alice in Wonderland. The book is lavishly illustrated and should delight curious children. Fear not ... it is not a boring series of equations. The math ideas are presented in a playful manner, with silly names for some of the more technical topics. Even a child in elementary school should find this book fairly easy to read. I was greatly pleased with the respect shown for the math and the level of sophistication of the ideas.

| A-E | F-J |
| K-O | P-T |
| U-Z |

Top

Once Upon A Number: The Hidden Mathematical Logic of Stories by John Allen Paulos. New York: Basic Books, 1998.

This book turned out to be minor disappointment because I was expecting something different and because I have read better works by the author. I was expecting the author to look at a broader topic - that of how story telling has many characteristics that are mathematical. In addition to logic there is the quality of symmetry in terms of plot and character development (something that had struck me in reading Shakespeare). But the focus did not expand from the theme of the subheading. Still, Paulos is a first rate author with an exceptionally accessible style. It will fascinate logicians far more than those looking (as I was) for a more global discussion of math and literature.
 

Poetry of the Universe: A Mathematical Exploration of the Cosmos by Robert Osserman. New York: Anchor Books, 1995.

Here is another book on the mathematics involved in cosmology. I'd rate this as a good read, but fairly dry. Definitely not as awe inspiring as a Carl Sagan book. If you cannot get enough on the subject, it has value - but it is not the best starting point.

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire. Washington, D.C.: Joseph Henry Press, 2003.

This work is about the Riemann Zeta Hypothesis, which states that all non-trivial zeros of the zeta function have real part one-half. The author does, for the most part, an admiral job of trying to break it down to novice level, but unfortunately the hypothesis is so complicated that I found myself skimming through much of the math in the latter half after he had lost me. I am sure I could take an entire college course on just understanding what the hypothesis says, and still come away without fully understanding it. However, the book is a fascinating look at how the mathematical community works. In this, it is much similar to the excellent PBS-Nova special Proof, about the solving of Fermat's Last Theorem.
 

Principles and Standards for School Mathematics by the National Council of Teachers of Mathematics. Reston, VA: NCTM, 2000.

The Principles and Standards issued by the NCTM are the guidelines for school systems nationwide to tailor their local goals for mathematics education. My first encounter with the Standards was as an undergraduate, where the book (the 1989 edition) was the required text for a course on teaching mathematics at Westfield State College. The 2000 Standards are greatly revised from that edition. The Standards are THE guide that all mathematics teachers. They propose a rigorous math education should be provided for all K-12 students and detail the means by which this can be achieved. This is must reading for all in the mathematics education field.

Probability 1: Why There Must Be Intelligent Life in the Universe by Amir D. Aczel. New York: Harcourt Brace & Co, 1998.

I was attracted to this book due to my interest in SETI (the Search for Extra-Terrestrial Intelligence). The author took the famous Drake Equation and explored it from a mathematical angle, trying to determine the probability that there may be at least one other intelligent source of life in the universe. I found it to be a wonderful application of probability. Students interested in the subject should find it fairly accessable. [See the author's other work on this list.] An enjoyable blend of math and space studies.

Proof A Play by David Auburn. New York: Faber and Faber, 2001.

"Proof" is a play that started its Broadway run in the Autumn of 2000 (and is currently scheduled to close in June 2001). The central issue of the play is a young woman's fear that she will lose her mind to insanity, just as her mathematician father slid away into madness. Her father had been a legendary mathematician whose creative energy burnt out early in life and then madness slowly took over. Catherine, the young woman, is a promising mathematician in her own right and fears that she will one day lose her mind as well. The play is an excellent read. [And the Broadway play turned was even better than the play! I was lucky to get a chance to see it in July 2001, with Toni Award winner Mary-Louise Parker in the lead role. It was a fabulous experience.] It is intelligent and gripping. The subject is far from totally fictional. Several great mathematician battled severe cases of mental illness in their later years, particularly Georg Cantor. I found the treatment of the subject to have a great level of tenderness.

 Q.E.D.: Beauty in Mathematical Proof by Burkard Polster. New York: Walker and Company, 2004.

This is a slim little volume, more of a gift or novelty item than a "real book," but it is quite charming. The book presents several of the most elegant proofs in very brief form, often relying on wonderful illustrations by the author. Much of it reminded me of the concept of "proofs without words," an interesting challenge and concept. Sometimes this approach was successful, other times the proofs begged for further explanation. Still, the book is a nice introduction to many of these proofs.

Science Matters: Achieving Scientific Literacy by Robert M. Hazen & James Trefil. New York: Anchor Books, 1990

Another non-math book, but it is related to math. I grabbed this book to brush up on science topics to prepare for the second Physical Science course required in my MSNS degree program. That ultimately got switched to a Psychology course -- but this still was a fun read. I have always enjoyed science, and this was a very concise, practical guide. Strongly recommended for college bound students.

Sphereland: A Fantasy About Curved Spaces and an Expanding Universe by Dionys Burger, translated from the Dutch by Cornelie J. Rheinbolt. New York: HarperPerennial, 1994 edition (original © 1965).

Sphereland is a sequel, of sorts, to the infamous Flatland. Where Flatland introduced readers to the possibilities of higher dimensions through the fictional adventures of a creature from a 2-dimensional universe, Sphereland explores the possibilities of "curved-space." The author takes up the story left off by Flatland author Edwin Abbott by having decendents of Abbott's characters discover that their 2-dimensional universe is curved through a third dimension. The tale serves as an introduction to the modern understanding of the Einsteinian universe, with curved space-time and an expanding universe. That said, I found this work to be inferior to Abbott's Flatland. Flatland had a far greater impact on me, really opening my mind to the possibilities of physical dimensions beyond our conventional three. When I then read works on cosmology and the shape of the universe, I accepted the ideas presented as distinctly possible. Sphereland did not strike me as radical. Still, it was an interesting read and has value.
 

The Story of Mathematics by Richard Mankiewicz. Princeton, New Jersey: Princeton University Press, 2000

If you are looking for a solid introduction to the history of mathematics, look at the next work, The Story of Mathematics by Motz & Weaver. Mankiewicz's work may be a nice "coffee table" book, rich with lavish color illustrations and full page plates, but it is mathematically poor. This work appears to have been influenced by the thinking that for each line of "mathematics" the sales will be cut in half. Thus, the author follows the development of mathematics from ancient times to the present with great attention to the contributions of many cultures, but the reader sees little of the mathematics. I was highly disappointed.
 

The Story of Mathematics by Lloyd Motz and Jefferson Hane Weaver. New York: Avon Books, 1993 (Trade Paperback edition, 1995)

This is another book that I was directed to through my MS degree program at RPI. The "culprit" this time was the Discrete Mathematics course (the professor for this class was Una Bray of Skidmore College). Much of the professor's "side bar" topics were out of the history of mathematics, and these topics were exceptionally fascinating. Thus, I made a mental note to look into math history books and I hope to someday take a course in this area. Concerning this book that I acquired at the Borders Books near RPI, I did enjoy it but it really needed to be retitled: The Story of the Mathematics Necessary for Use in Physics. This was a follow-up work to the authors' The Story of Physics and they did not in any way disguise that physics is their first love. While I am all in favor of finding connections between mathematics and its applications, this work was definitely biased in that nearly everything was seen in terms of how it related to physics. Since the link is strong, however, this can be somewhat forgiven, especially as the book is contains many engaging stories and is packed with information. Readers who have some college math background will find no difficulty with the book and it should also provide a nice source for further research. High school students will find much of the modern math to be well beyond their experience, but those sections can be skipped over without much loss, hopefully to be returned to later in their studies.
 

To Infinity and Beyond: A Cultural History of the Infinite by Eli Maor. Publisher: Princeton, NJ: Princeton University Press, 1991. [© 1987]

This book explores the concept of infinity from four distinct perspectives: mathematical, geometric, aesthetic, and cosmological. For me the most interesting portion was the mathematical treatment, largely due to my just having completed Introductory Analysis at the time I read it (late December, 2000). I suspect this portion of the text would be the most difficult for those who have yet to take some of the higher mathematics courses. The geometric treatment was interesting, bringing together several topics such as Euclid's Fifth Postulate and non-Euclidean geometries. The aesthetic section is dominated by works by M. C. Escher. This is good on the one hand since Escher is the master of the infinite in art, but a wider spectrum would have been nice. The cosmological treatment deals with our understanding of the vast scope of the cosmos, from ancient times to modern, with the scale expanding as we gained better understanding of our universe. This section would make a nice introduction for those interested in exploring further. On the whole, a wonderful book, with many illustrations and quotes on the subject of the infinite.
 

A Tour of the Calculus by David Berlinski. New York: Pantheon Books, 1995.

Another great math read! I picked this up the summer before I was to teach calculus for the first time, and I think it really helped me understand the subject better. Any calculus student will find the effort well worth it. The author combines the talents of a fabulous mathematics teacher with those of a master story teller. This one ranks up there with William Dunham's Journey Through Genius and J. A. Paulos' Innumeracy as absolute must reads! [This book also was "revisited" when I took Introductory Analysis and this book was part of the required readings.]

| A-E | F-J |
| K-O | P-T |
| U-Z |

Top

Uncle Petros & Goldbach's Conjecture by Apostoios Doxiadis. New York: Bloomsbury, 2000 (English translation. Originally published in Greek, 1992)

Absolutely charming! This is a must read novel for anyone who loves mathematics. Yes, it is indeed a novel, a rarity on this list. The story is a bittersweet tale of a young, brilliant mathematician who sets out to prove Goldbach's Conjecture, spending an entire lifetime in the pursuit of this mathematical "holy grail." (The conjecture, still unproven, claims: All even numbers, greater than two, can be expressed as the sum of two primes.) This work captures the passion we in mathematics have for the subject. I easily saw some of myself in the character of the nephew, who is not the "great mathematician" but through whom the story is told. The nephew has ambitions of becoming a mathematician, sees that he does not have the gift for it and is forced to turn aside, but is able to experience some of the beauty of mathematics through his entry level studies and through his relationship with his Uncle Petros. The novel is both smart ... the author being both a professional mathematician and filmmaker (according to the bio on the jacket) ... and passionate. I put this on a par with the Broadway play Proof as first class unions of mathematics and human drama.
 

The Universe and the Teacup: The Mathematics of Truth and Beauty by K. C. Cole. San Diego, CA: Harcourt Brace & Company, 1997 (paperback edition, 1999)

I was very eager to read this book once I learned the author was female. Gender inequity is a major issue in mathematics and math education. You will note that the bulk of the works cited in this web page were the product of male authors, so I was very pleased to be able to add a work by a female author to begin to establish a measure of balance. That said, I, alas do not rank this book very high. The approach was similar to that of John Allen Paulos in Innumeracy. The author sought to demonstrate how mathematics permeates society and that math is key to understanding the world around us. In that she was fairly successful. However, the topics were not terribly well organized. The topics flowed like a stream of consciousness, nearly haphazard at times, jumping here and there and then repeating far too often. While the basic style was readable, I would have preferred more thorough investigation of topics before switching to a new connection. Perhaps if I had read this before other similar works I would have enjoyed it more.

Visions: How Science Will Revolutionize the 21st Century by Michio Kaku. New York: Doubleday, 1997

In this book the author looked forward to what he saw as important areas of research or technological breakthroughs in the new millennium. Some of it was interesting, but, again, he did not succeed in recapturing the excitement and wonder of Hyperspace.

What is Mathematics? An Elementary Approach to Ideas and Methods (Second Edition) by Richard Courant and Herbert Robbins (Revised by Ian Stewart). New York: Oxford University Press, 1996.

I saw this book at a Barnes & Noble and thought it would be a valuable addition to my math library. Some time later, I mentioned it in one of my grad school courses, and the professor remarked that this book revitalized mathematics instruction in this country. Apparently, I made a good choice. The book is a tough read, but if you are planning a career in math you really should give it a shot. Definitely written for a serious math student, this will be largely inaccessible to students just starting out their math studies. Keep it in mind until you have a few higher math courses under your belt. [For more on Ian Stewart, click here]

| A-E | F-J |
| K-O | P-T |
| U-Z |

Top

Why Numbers Count: Quantitative Literacy for Tomorrow's America Edited by Lynn Arthur Steen. New York: College Entrance Examination Board, 1997.

This is another book in my quest to become better informed on the subject of how our nation fares in terms of mathematical / quantitative literacy ... and how we can improve in that regard. [See John Allen Paulos] This volume I obtained via the National Council of Teachers of Mathematics. It is comprised of a series of essays on the topic, with authors coming from various backgrounds. Too often "we" math teachers never look beyond the classroom for what mathematics is important for today's students. This book looks at those mathematics skills judged necessary by not only mathematicians and math educators but also by those in the work place. I found several of the essays to be highly informative and believe this book will make a nice resource for potential mathematics teachers. It would have fit nicely into several of the college level courses I took.

Zero: The Biography of a Dangerous Idea by Charles Seife, Drawings by Matt Zimet. New York: Viking, 2000.

A book about the number zero??? You must be joking! It's no joke ... and it is a terrific book. As a mathematics teacher, I see every day how important the number zero is and how interesting its cultural history is. Which cultures first started using it? When did it come over to the West? It makes a great topic for discussing multicultural issues in math class. This book will make a great source for ideas and information. The author charts the history of zero, starting off with how it impacted the "millennium" debate. Since the Romans had no number zero, early calendars started with year one. The discussion then turns to how the Roman numbering system fell away and the Hindu-Arabic system took its place. The reason was largely due to the efficiency of the system. The author then turned his attention to how zero impacted the development of calculus, where many of the problems center on something approaching zero. Attention was also paid to to the compliment of zero ... infinity. Thus, the book explored much of the foundation of modern Analysis. Rounding out the work is an exploration of zero in nature, be it absolute zero on the temperature scale or how the universe arose out of zero volume in the Big Bang. Fascinating stuff! Highly recommended for readers from high school age and up.